Elegant proofs 4 – The optimum shape of an arch

Large span arch structures have been constructed for well over two thousand years, but the first recorded analytical treatment was given by Robert Hooke in 1675 who at the end of his treatise on helioscopes added the following statement “to fill up the vacancy”:

“The true mathematical and mechanical form of all manner of arches for building, with the true butment necessary to each of them. A problem which no architectonick writer hath ever yet attemted, much less performed. abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuvx” *

The solution to this anagram (no doubt written as such to annoy Newton) was published in 1705, after Hooke’s death as:
“Ut pendet continuum flexile, sic stabit contiguum rigidum inversum”
which translates as:
“As hangs the flexible line, so but inverted will stand the rigid arch”

In other words, the shape of a flexible line, a catenary, which is in pure tension, will if inverted be in pure compression, and hence the ideal shape for an arch. The same realisation was apparently used by the builders of the The Roof of the Taq-i-Kisra some 1000 years earlier, but in general the shape of large span arch structures is not a catenary, circular or elliptical shapes being almost universally used for masonry arch bridges, and parabolic shapes for free standing arches.  It is often suggested that these alternative shapes are used as convenient approximations to a catenary, but in fact there are two additional factors which change the optimum shape for arch structures:

  1. The catenary is the optimum shape for a free standing arch of constant cross section, but in a typical arch bridge the majority of the weight will be in the bridge deck, and the optimum supporting shape will tend towards a parabola.
  2. For arch structures in which the roadway is supported on fill (such as typical masonry bridges) the fill applies horizontal loads to the arch, as well as vertical, and the optimum supporting shape will tend towards a circular or elliptical arc for very high fills, or some more complex shape for shallower fills.
Arch bridge at Totnes, Devon

Arch bridge at Totnes, Devon

More details can be found at: Arch Structures

* Interestingly, all transcriptions of the anagram I have found on the Internet insert an additional e, but the reproduction of the original publication here clearly shows that there are only 5 e’s, and this is consistent with the solution of the anagram.  This appears to be an example of Stephen Jay Gould’s “Fox-terrier fallacy” where an erroneous statement becomes accepted through repeated copying.

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10 Responses to Elegant proofs 4 – The optimum shape of an arch

  1. xm carreira says:

    Nice post and pictures. Unfortunaltely the gothic arch was also wrongly used in some bridges of the XIV and XV centuries instead of the circular/elliptical and parabolic arches.


  2. Ed B says:

    Excellent post … and pictures ! I rely on the circular arch for the structures I build for the yard … although, they are purely decorative.


  3. dougaj4 says:

    Ed – had a look at your blog; very nice.

    I like arches🙂

    Did you see the post about the roof of the Taq-i-Kisra?



  4. Ed B. says:

    Thanks for the look …. I’m rather partial to arches myself !!
    I have wondered how I would go about building molds for an arch shaped in that fashion: instead of one or two I would need at least three … and each one would be different. Think I’ll stick to the half-round or circular arch … Depending on how many segments will comprise the arch I can make one or two identical molds without much problem.


  5. George Carty says:

    Your link to the Arch Structures PDF is broken — the correct link is here.


  6. dougaj4 says:

    Thanks George. I have corrected the link in the text as well.


  7. Pingback: Arches, anagrams and plagiarism « Newton Excel Bach, not (just) an Excel Blog

  8. Bill Harvey says:

    Of course, arch bridges in the modern world have to sustain much bigger live loads. Railway axles have gone from 6 tons to 25 and more since the original railway bridges were built. Live load is now significant, so there is no perfect shape for any arch.



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